Calculus - Integrals Projects

(a) Find the second derivatives of f(x)=∫_-∞^x^2 / 2 e^x-t^2 / 2 d t and g(x)=∫_-∞^x^2

(a) Find the second derivatives of $f(x)=\int_{-\infty}^{x^{2} / 2} e^{x-t^{2} / 2} d t$ and $g(x)=\int_{-\infty}^{x^{2} / 2} e^{-\left(x^{2}+1\right) t^{2}} d t$.

(b) Derive the solution of the ordinary differential equation

\[\frac{d^{2} y}{d x^{2}}=f(x), \quad x>0, \quad y(0)=0, \quad \frac{d y}{d x}(0)=0\]

in the form

\[y(x)=\int_{0}^{x}(x-t) f(t) d t\]

Problem 2: Consider the partial differential equation

\[\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}\]

By assuming a solution of the form $u(x, t)=X(x) T(t)$, deduce that

\[u_{\alpha}(x, t)=\left(A_{\alpha} \cos (\alpha x)+B_{\alpha} \sin (\alpha x)\right) e^{-\alpha^{2} t}\]

where $A_{\alpha}$ and $B_{\alpha}$ are constants, is a solution for any constant $\alpha$. Show that if we now impose the boundary conditions $u(0, t)=0, u(\pi, t)=0$, then this reduces the possible solutions to those of the form

\[u_{n}(x, t)=B_{n} \sin (n x) e^{-n^{2} t}\]

for $n=0,1,2, \ldots$ and where $B_{n}$ is a constant. Hence or otherwise find the solution of the

problem

\[\begin{aligned}

& \frac{\partial u}{\partial t}=\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}},\quad u(0,t)=0,\quad u(\pi ,t)=0 \\

& u(x,0)={{\sin }^{2}}(x),\quad 0<x<\pi \\

\end{aligned}\]

Problem 3: By considering $S_{n}=1+x+x^{2}+\cdots+x^{n}$, or otherwise, show that for $|x|<1$

\[\frac{1}{1-x}=1+x+x^{2}+x^{3}+x^{4}+x^{5}+\cdots\]

Use this result to find the Taylor series of the functions below and indicate the values of $x$ for which the corresponding series converges:

\[\text { (a) } \frac{1}{2-x}, \quad \text { (b) } \frac{x}{1+x-2 x^{2}}, \quad \text { (c) } \log \left(\frac{1+x}{1-x}\right) \text { . }\]

Problem 4:

For the following matrix, find all eigenvalues and all (normalised) eigenvectors:

\[\left(\begin{array}{ccc}

6 & -2 & 2 \\

-2 & 5 & 0 \\

2 & 0 & 7

\end{array}\right)\]

Problem 5: Solve the following ordinary differential equation initial value problems for $y(x)$

\[{{y}^{\prime }}+xy=0,\quad y(0)=1\text{ , }\]
$x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=6, \quad y(1)=0, y^{\prime}(1)=1$,
$y^{\prime \prime}+y^{\prime}-6 y=1, \quad y(0)=0, \quad y^{\prime}(0)=2$

Problem 6: For the following functions, find the critical points, determine their nature (maxima, minima, inflection, etc.) and sketch the graph or surface:

$f(x)=\frac{x}{1+x^{2}}$ for $x \neq 0$,
$\quad g(x)=|x| e^{-|x-1|}$,
$\quad F(x, y)=\left(x^{2}-4\right)^{2}+y^{2}$,
$\quad G(x, y)=\sin x \sin y \sin (x+y) \quad(0 \leq x, y \leq \pi)$.

Problem 7:

Consider the heat equation

\[\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}\]

Show that if $u(x, t)=t^{\alpha} \phi(\xi)$ where $\xi=x / \sqrt{t}$ and $\alpha$ is a constant, then $\phi(\xi)$ satisfies the ordinary differential equation

\[\alpha \phi-\frac{1}{2} \xi \phi^{\prime}=\phi^{\prime \prime}\]

(where $\prime \equiv d/d\xi $ ). Show that

\[\int_{-\infty}^{\infty} u(x, t) d x=\int_{-\infty}^{\infty} t^{\alpha} \phi(\xi) d x\]

is independent of $t$ only if $\alpha=-\frac{1}{2}$. Further, show that if $\alpha=-\frac{1}{2}$ then

\[C-\frac{1}{2} \xi \phi=\phi^{\prime}\]

where $C$ is an arbitrary constant. From this last ordinary differential equation, and assuming $C=0$, deduce that

\[u(x, t)=\frac{A}{\sqrt{t}} e^{-x^{2} / 4 t}\]

is a solution of the heat equation (here $A$ is an arbitrary constant).

Show that as $t$ tends to zero from above,

\[\lim _{t \rightarrow 0+} \frac{1}{\sqrt{t}} e^{-x^{2} / 4 t}=0 \text { for } x \neq 0\]

and that for all $t>0$

\[\int_{-\infty}^{\infty} \frac{1}{\sqrt{t}} e^{-x^{2} / 4 t} d x=B\]

where $B$ is a (finite) constant. Given that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$, find $B$.

What physical and/or probabilistic interpretation might one give to this solution $u(x, t)$ ?

Problem 8: Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent, identically distributed random variables, each having a distribution function $F_{X}(x)$. Let $M=\min \left\{X_{1}, X_{2}, \ldots, X_{n}\right\} .$ Find the distribution function of $M$. Now suppose $F_{X}$ is the uniform distribution over $(0,1)$. What is the probability density function of $M$ ?

10. Let $X$ and $Y$ be random variables with joint probability density

\[f_{X, Y}(x, y)=\left\{\begin{array}{cl}

c e^{-x-y}, & 0<x<y<\infty \\

0 & \text { otherwise. }

\end{array}\right.\]

Determine the value of the constant $c$. Determine whether $X$ and $Y$ are independent.

11. A gambler plays a game in which there is a probability $p$ of winning one unit and probability $q=1-p$ of losing one unit. Successive plays of the game are independent. What is the probability that, starting with $x>0$ units, the gambler's fortune will reach N? If $p=q$ find the expected time for a gambler who starts with $x>0$ units to lose them all.

12. Let $X$ and $Y$ have a bivariate normal distribution.

Show that $X$ and $Y$ are normally distributed, and find their mean and variance.
Show that $X+Y$ is normally distributed, and find its mean and variance.

Problem 13: The random variable $Y$ has the standard normal density with mean 0 and variance 1 , $Y \sim \mathcal{N}(0,1)$. Find the distribution and density functions of $V=Y^{2}$

The moment generating function $M_{X}(t)$ of a random variable $X$ is defined by $M_{X}(t)=$ $E\left[e^{t X}\right] .$ If $X \sim \mathcal{N}(0,1)$, show that $M_{X}(t)=e^{t^{2} / 2}$

Let $X$ and $Y$ be independent standard normal random variables, and $Z=X Y$. Find $M_{Z}(t)$, either by direct calculation or via the tower law in the form $E\left[e^{t Z}\right]=$ $E\left[E\left[e^{t Z} \mid Y\right]\right]$

Problem 14: A hen lays $N$ eggs, where $N$ has the Poisson distribution with parameter $\lambda$. Each egg hatches with probability $p$ independently of the other eggs. Let $K$ be the number of chicks.